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U.S. DEPARTMENT OF AGRICULTURE • FOREST SERVICE • FOREST PRODUCTS LABORATORY • MADISON, WI

U.S. FOREST SERVICERESEARCH NOTE

FPL-059September 1964

BENDING STRENGTH AND STIFFNESS OF PLYWOOD1

By

Forest Products Laboratory, Forest ServiceU.S. Department ofAgriculture

Summary

Layers of a plywood beam in which the grain direction is parallel to the spanhave properties distinctly different from those of adjacent layers because of the90° orientation of adjacent plies. Such differences in properties in adjacent pliescomplicate the stress distribution at various points in the cross section, as com-pared to material with the same properties at all points in the cross section. Theproperties of a plywood beam thus depend on the properties in the span directionof each ply and on the construction of the plywood--thatis, on the thickness andnumber of plies.

This report presents data from tests of plywood of various ply thicknesses andnumbers. Methods are given for computing the strength and elastic propertiesfrom the construction of the plywood and the properties of the individual plies.

Introduction

Wood is anisotropic in character, having three mutually perpendicular axes ofsymmetry--longitudinal, radial, and tangential. The properties of wood in the

1This Note is a slight revision of Forest Products Laboratory Report No. 1304, originally writtenin 1942 by Alan D. Freas under the title, "Method of Computing the Strength and Stiffness ofPlywood Strips in Bending." It was first revised in 1946 and then revised again in 1956 underthe present title.

longitudinal direction are found to be distinctly different from correspondingproperties in the other two directions. Bending strength in the longitudinal direc-tion, for example, is found to be some 15 to 20 times as great as in the tangen-tial direction. Other properties also show differences.

Plywood, consisting as it does of alternate layers with grain directions atright angles, introduces further complications. The direction of principal stressin a plywood beam is parallel to the span, Thus the principal stresses are alter-nately parallel to grain and perpendicular to grain from layer to layer. The dif-ferences in strength and modulus of elasticity in the parallel-to-grain andacross-the-grain directions thus complicate the stress distribution at variouspoints in the cross section of a plywood beam as compared to a beam of materialwith the same properties at all points in the cross section.

The objective of this report is to develop methods of calculating the elasticand strength properties of plywood beams from the properties of the componentplies and the construction.

Scope of Study

The bending tests included in this investigation of plywood covered variousspecies, thicknesses, and constructions. Specifically, the variables consideredwere:

Three species--Douglas-fir, Sitka spruce, and yellow-poplar. Five veneer or ply thicknessess--1/24, 1/16, 1/12, 1/8, and 3/16 inch. Five combinations with respect to number of plies--1, 3, 5, 7, and 9. Five arrangements of plies--2

(a) Single-ply veneer with grain parallel to span.(b) Laminated wood with grain of all plies parallel to span.(c) Laminated wood with grain of all plies perpendicular to span.(d) Plywood with grain of outer plies parallel to span, grain of

adjacent plies at right angles.(e) Plywood with grain of outer plies perpendicular to span,

grain of adjacent plies at right angles.Related tests--standard specimens of solid wood with grain parallel

to span.

Tests were made only on specimens in which all plies were of the same spe-cies and the same nominal thickness. Similarly, no defects were permitted in the2

It is desirable, for convenience, to employ a more concise, though less accurate, terminology forthe various ply arrangements. The following term will, therefore, be used in later portions ofthe report: “All-parallel” will be used to mean ply arrangement described under (b) above: “all-perpendicular,” that under (c); “outer-parallel,” that under (d): and “outer-perpendicular,”thatunder (e).

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specimens tested. Defects such as knots and spiral grain that. occurred in asheet of veneer were eliminated from the material that was to form the speci-men. Deviation of the grain from the plane of the veneer (diagonal grain) is to beexpected in rotary-cut veneer as a result of unavoidable eccentricity of thegrowth rings. No attempt was made to eliminate this defect nor to evaluate itseffect, because considerable variation is possible within even a small area andbecause of the difficulty of measuring its magnitude. The effect of defects, there-fore, has been effectively eliminated in this study.

Bending tests were made on 2- by 2- by 30-inch solid specimens cut from thecores of the peeler blocks. Ten specimens were cut from each of two cores forboth Douglas-fir and Sitka spruce--a total of 20 specimens for each species.Yellow-poplar cores were not available for these tests.

The following tests were made on veneer, plywood, and laminated specimens:

Sitka Spruce

Three-ply.--Four tests were made on each of four veneers of each thickness(1/24, 1/16, 1/12, 1/8, 3/16 inch) from each of two trees and of each of the fivebasic constructions. The total number of tests was 800.

Five-ply.--The5-ply test series was the same as the 3-ply series, exceptthat two tests were made on panels from each of two sheets from each of twotrees for each thickness and construction. The total number of tests was 200.

Seven-ply.--The 7-ply test series was the same as the 3-ply series, exceptthat four tests were made on panels from each of two trees for each thicknessand construction. The total number of tests was 200.

Nine-ply.--The 9-ply test series was the same as the 7-ply series, with thesame total of 200 tests.

In all, 1,400 tests were made on Sitka spruce panels.

Douglas-fir

The series of tests made on Sitka sprucepanels was also made on Douglas-firpanels. The total number of tests was also 1,400.

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Yellow-poplar

Three-ply.--The3-ply yellow-poplar was given the same series of tests asthe 3-ply Sitka spruce. The total number of tests was 800.

Five-ply.--The5-ply test series was the same as the 3-ply series, exceptthat one test was made on panels from each of two sheets from one tree for eachthickness and construction. The total number of tests was 50.

Seven-ply.--The7-ply test series was the same as the 5-ply series. The totalnumber of tests was also 50.

Nine-ply.--The 9-ply test series was the same as the 7-ply series, except thatone test was made on panels from one sheet from one tree for each thickness andconstruction. The total number of tests was 25.

The total number of tests on yellow-poplar panels was 925.

Total

In all, 3,725 tests were made on the panels of all three species. In addition,bending tests of 2- by 2- by 30-inch specimens were made by standard methodson 20 Sitka spruce and 17 Douglas-fir specimens cut from the cores of the peelerblocks.

Material and Fabrication

Material

The exact region of growth of the material is not known. The Sitka spruce andDouglas-fir veneer were cut at Tacoma, Wash., and the yellow-poplar veneerwas cut at Huntington, W. Va.

For each species, the veneer was cut by the rotary process from 8-foot boltsof 2 logs into 4-foot-wide sheets, except occasionally where defects made itnecessary to cut smaller widths. Veneers of each thickness were cut from eachlog. The veneer sheets of each thickness were numbered successively as theywere cut and were identified by log and sheet numbers.

The veneer was dried at the mills by usual methods. The cores remainingafter the Sitka spruce and Douglas-fir veneer had been cut from the peeler blocks

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were left intact and were shipped inthegreen condition to the Laboratory, wherethey were cut into 3-1/2-inch-thickplanks that weresubsequently air dried. Thecores from the yellow-poplar were not obtained. The core diameters were: Sitkaspruce log 1, 56 inches: Sitka spruce log 2, 45-1/2 inches; Douglas-fir log 1,32 inches; Douglas-fir log 2, 34 inches.

The veneer was specially cut for this study and was of high quality with onlyvery minor defects.

Matching and Marking of Specimens

The specimens were made up in matched sets of five specimens. Each setconsisted of one specimen of single-ply veneer and one specimen each of theall-parallel, all-perpendicular, outer-parallel, and outer-perpendicular plyarrangements,2 all with the same number of plies and made of plies of the samethickness.

Since the chief function of the tests was to develop methods of computing thestrength and stiffness of plywood, it was desirable to reduce, as far as possible,the effect of variability of material. For this reason, all material for a given setof specimens was cut from a single sheet of veneer.

Tests were made generally on not less than four sets of specimens from eachtree. Usually, each set was from a different sheet of veneer, but for the smallernumber of plies and the thinner veneers it was sometimes possible to get all thematerial for four sets of tests from a single sheet of veneer. In such cases, morethan four sets of specimens were tested.

Gluing and Conditioning Specimens

The veneers were bonded with casein glue (Forest Products Laboratory for-mula 4b) after they had been conditioned to approximately constant weight in anatmosphere maintained at 72° F. and 52 percent, relative humidity (conditionsthat produce approximately 10 percent equilibrium moisture content in wood).After the specimens were glued, they were stored under the same conditions untiltheir weight was about constant before they were tested. The gluing was done atroom temperature with a spread of about 0.032 pound of glue per square foot ofjoint area.

The solid specimens were conditioned at 72°F. and 52 percent relative humidityuntil their weight was approximately constant before they were tested.

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Methods of Test

All bending specimens were tested under center loading. The specimens were2 inches wide and were made 2 inches longer than the span to provide a 1-inchoverhang at each end. The span used varied with the different constructions so asto afford a uniform effect of shear deformation. The span to be used for eachconstruction was computed from the span-depth ratios based on the nominaldepth of the specimen. For the all-parallel, outer-parallel, and single-ply con-structions, the span-depth ratio was 48. For the all-perpendicular and outer-perpendicular constructions, the span-depth ratio was 24.

The radius of curvature of the loading block was made 1.5 times the nominaldepth of the specimen. The rate of descent of the movable head was determined

3by the equation (1)

where N is the rate of descent of the movable head in inches per minute, z therate of outer fiber strain (taken here to be 0.0015 inch per inch per minute), Lthe span in inches, and d the depth of the beam in inches.

The specimens were supported on knife edges that were adjustable laterally tocompensate, where necessary, for warp. Roller supports were provided betweenthe knife edges and the specimen to eliminate end restraint. Figure 1 shows de-tails of the adjustable knife edges and the roller supports.

Simultaneous readings of load and deflection of the center with respect to theends were obtained throughout each test. Deflections during the early portion ofthe test were measured by means of a dial gage graduated to 0.001 inch. For thelonger specimens, the dial gage was attached to a metal yoke supported on nailsdriven into the specimen at the neutral plane immediately above the supports. Thespindle of the dial gage was attached to a nail driven into the neutral plane at thecenter of the specimen, as illustrated in figure 1.

With the shorter specimens, it was necessary to attach the dial gage to the baseof the support, with the spindle of the dial gage supported from a yoke attachedto the center of the specimen (figs. 2 and 3).

3 Underlined numbers in parentheses refer to literature cited at the end of the text.

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For tests of single-ply veneer, it was impractical to make any attachments tothe specimen itself and, in such tests, the dial gage was attached to the base ofthe support and the spindle of the dial gage was allowed to bear against thebottom of the specimen, directly below the loading block (fig. 4).

For the longer specimens, where deflections in excess of the capacity of thedial gage were encountered, a steel scale graduated to 0.01 inch was suspendedfrom the center of the specimen and a fine wire was stretched between nailsdriven into the specimen directly above the supports. The deflection was meas-ured by observing, through a telescope, the passage of the scale past the wire.The tests were discontinued when it was certainthat the maximum load had beenpassed.

Where loads were small and the testing machine was not sensitive enough togive satisfactory results, a more accurate weighing system, employing a plat-form scale, was used. The platform scale was supported on the stationary plat-form of the testing machine, and motion was imparted to the loading block bymeans of rods passing through holes in the scale platform and connecting themovable head of the testing machine with the loading block. The testing machinewas thus used only as a means of achieving the desired rate of movement of theloading block, while loads were measured with the platform scale to 0.01 pound.This device is illustrated in figure 3.

The solid 2- by 2- by 30-inch specimens were tested under center loading ona 28-inch span at a head speed of 0.10 inch per minute (2). The loading block hada radius of 3 inches. Deflection of thecenter with respect to the ends was meas-ured with a device similar to that shown in figure 1.

All tests were made in an atmosphere maintained at 72°F. and 52 percentrelative humidity.

Moisture determinations were made on pieces that were cut from each speci-men immediately after it was tested.

Discussion of Results

The analyses discussed in this section are based on a comparison of theoret-ical values calculated from strength properties determined from tests on lami-nated and plywood specimens. In each case, the laminated specimen consideredhad the same number and thickness of plies as the plywood specimen with whichcomparisons were made. This procedure was adopted to eliminate any possible

FPL-059 -7-

effect from differences in depth of beam or from differences in the number ofglue joints. The results from tests of veneer were not used in the analyses.

No corrections of data were made for the large deflections encountered duringtest. These corrections represent a negligible part of results, thus were notconsidered.

Modulus of Elasticity

Theoretical considerations show (7, 11) that the deflection of points on thecentral line of a centrally loaded rectangular strip of plywood is given by

(1)

where w is the deflection; P the central load; a the span; E the apparent modu-clus of elasticity in bending; I the moment of inertia, based on the full cross sec-tion of the specimen; µ

LTthe Poisson’s ratio associated with the tangential di-

rection and stress in the longitudinal direction; µTL

the Poisson’s ratio associated

with contraction in the longitudinal direction and stress in the tangential direc-tion; B and e factors involving elastic constants of wood; and h the depth of thespecimen,

The factor (1 - µLT

µTL

) B accounts for the difference in deflection at dif-

ferent points across the width of the specimenthat results from anticlassic cur-vature. The difference between this factor and unity was found to be relativelyunimportant for the ratios of span to width used in these experiments. It wasabout 0.99 for all types of specimens except those with the grain of all plies per-pendicular to the span, for which it was found to be 1.00.

h2

The factor (1 + e 2) accounts for the deflections resulting from shear de-a

formations. This factor varied in its effect with different constructions fromabout 1.01 to about 1.13.

The modulus of elasticity is different at different points in a cross section ofa plywood strip, since the transverse plies have a modulus of elasticity parallel

FPL-059 -8-

to the span that is only about 3 to 5 percent of that of the longitudinal plies. Theterm “apparent modulus of elasticity in bending” is therefore used to mean acomposite or overall modulus such as might be obtained by the use of the usualequations with tension or compression data or withstatic bending data correctedfor shear deformation. The apparent modulus ofelasticityin bending, E , has thegeneral equation C

(2)

where EC

and I are as previously defined, Ei

is the modulus of elasticity of the

ith ply parallel to the span, I the moment of inertia of the ith ply about the cen-i

tral line of the full cross section, and n the number of plies. Where r is equalto E

T/E

L, E

1is the apparent modulus for a strip with the grain of the outer

plies parallel to the length of the specimen, and E2

is the apparent modulus for

a strip with the grain of the outer plies perpendicular to the length of the speci-men, and it is assumed that all plies are of the same thickness, as was the casein the tests made in this study, E may be shown to have the following values:c

Number Equation for Ecof

plies Outer-parallel Outer-perpendicular

r + 26 1 + 26r3 E

1=

27E

LE

2=

27 EL

26r + 99 26 + 99r=5 E1 125 EL

E2

=125

EL

(3)99r + 244 99 + 244r

7 E1

=343 EL

E2

=343 EL

244r + 485 244 + 485r=9 E1 729

EL

E2

=729

EL

EL

is the modulus of elasticity of wood in the longitudinal direction, and ET is

the modulus of elasticity of wood inthetangential direction. It may be noted that,

FPL-059 - 9 -

in equation (1). Ec

= EL

for specimens with the grain of all plies parallel to the

length of the specimens, and E = ET

for specimens with the grain of all pliesperpendicular to the length. c

Values of EL

and ET

were obtained by use of equation (1) from tests on all-

parallel and all-perpendicular specimens, respectively, and were used withequations (3) to calculate values of E and E Values of E and E and calcu-

1 2 L Tlated values of E and E are shown in tables 1, 2, and 3 in columns 5, 7, 8, and9, respectively.

1 2

Values of E1

and E2

were obtained by use of equation (1) from tests on outer-

parallel and outer-perpendicular specimens, respectively, and are shown intables 1, 2, and 3 in columns 11 and 13 as E

1(obs) and E

2(obs).

Throughout the report, the term “calculated value” is used to mean a valuepertaining to plywood derived from theoretical considerations based on the con-struction of the plywood and from values computed from tests of all-parallel andall-perpendicular specimens. The term “observed value” is used to mean a valuecomputed from tests on plywood. The observed values may then be comparedwith the calculated values to check the validity of the theoretical considerationsinvolved in setting up the formulas for the strength and stiffness of plywood.

The ratios of observed values of E1 and E2 to calculated values are shown in

columns 14 and 15 of tables 1, 2, and 3. The plywood specimens that gave theobserved values were matched with the laminated wood specimens that gave theresults used in obtaining the calculated values.

All values except specific gravity in tables 1, 2, and 3 have been adjusted toa uniform moisture content basis of 10 percent on the assumption that moistureadjustments for plywood and laminated wood maybe made in the same manner asfor solid wood (12). Other work (5) has confirmed the validity of this assumption.

From a consideration of tables 1, 2, and 3, it may be seen that there is appar-ently no uniform variation of the ratio of observed to calculated values of E 1 and

E2

, either with ply thickness for a given number of plies, with number of plies

for a given thickness, or between tree 1 and tree 2. It seems proper, therefore,to consider all of these ratios together for each species. Table 4 presentsstatistics summarizing the comparison between observed and calculated valuesgiven in detail in tables 1, 2, and 3.

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From table 4 it may be seen that, in general, the ratio between observed andcalculated values is approximately unity. Considering the magnitude of the stand-ard deviation it may be seen at once that none of the mean values of the ratio issignificantly different from unity.

From this it may be concluded that equation (1) represents, within the limitsof variability of the material, the relation between load and deflection of a cen-trally loaded rectangular strip of plywood for stresses up to the proportionallimit, with the values of apparent modulus of elasticity for the plywood defined byequation (2).

Fiber Stress at Proportional Limit

Theoretically, the stress at a point z distant from the neutral axis of the crosssection of a strip of plywood subjected to a bending moment M is given by

(4)

where f is the unit stress at a point z distant from the neutral axis, M the bendingmoment, z the distance from the neutral axis to the point being considered, E

the modulus of elasticity of the material in the direction of the stress at the pointbeing considered, E the apparent modulus of elasticity in bending for the stripcof plywood, and I the moment of inertia based on the full cross section of thespecimen.

Equation (4) applies, of course, only when stresses are below the proportionallimit at all points in the cross section. The analysis leading to equation (4) isbased on the assumption of linear variation of strain across the cross section,and is applicable only where the shear deformations are not great enough to in-validate this assumption, as in beams of relatively large span-depth ratio or inregions of low shear, such as the centralportion of a uniformly loaded beam.

The resisting moment of a plywood beam for a stress f in the extreme fiberis given by

(5)

where c is the distance from neutral axis to extreme fiber in inches.

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X

fEC

If E is replaced by fa, which may be called an apparent stress, equation (5)

xmay be replaced by

(6)

where

(7)

If the apparent stress at proportional limit is denoted by FL’

FT

, and F1’

for

all-parallel, all-perpendicular, and outer-parallel specimens, respectively, itmay be seen that, based on equation (7),

(8)

It may be noted that, for all-parallel and all-perpendicular specimens, the ratioE /E becomes unity, and equation (4) reduces to the usual equation for stressc xin a bent beam.

Values of FL

and FT

were obtained, by the use of equation (6), from data on

all-parallel and all-perpendicular specimens, respectively, and values of FL

were used with equation (8) to calculate values of F1. Values of FL and FT

and

F1 (calc.) are shown in tables 5, 6, and 7 in columns 5, 10, and 7, respectively.

Values of F1 were obtained by use of equation (6) from data on outer-parallel

specimens, and are shown in tables 5, 6, and 7 in column 15 as F1 (obs).

The observed and calculated values were then compared. The ratios of ob-served to calculated values are shown in column 20 of tables 5, 6, and 7 and aresummarized in table 8. The calculations were made for groups of specimens asdescribed in the discussion of modulus of elasticity.

It may be noted that the observed values were approximately 90 percent of thecalculated values; ranging from 0.72 to 1.09 times the calculated values. This is

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because the proportional. limit in the bending test is reached first at only onelocation in the beam. This location is directly under the load in the extremecompression fiber. As the test proceeds, adjacent material also reaches theproportional limit. A considerable amount of material must be affected beforethe load deflection curve is sufficiently influenced to indicate a proportionallimit (3). Thus the proportional limits obtained from bending tests are alwaysgreater than those obtained from compression tests.

The tensile and compressive strengths of wood perpendicular to the grain arevery low, Calculations indicate that the tensile strength of the outer transversetension ply of a plywood strip that has its face grain direction perpendicular tothe span is exceeded by the time the outermost longitudinal ply has reached itsproportional limit. When the tensile strength of the tension transverse ply hasbeen exceeded, it can no longer contribute to the bending strength of the plywoodstrip. Therefore, the proportional limit bending strength of a plywood strip iscontrolled by that of the outermost longitudinal ply, and the outer transverse plymay be neglected in computing the proportional limit strength of the strip.

There are indications that the proportional limit in compression perpendicularto grain is reached at a deformation about 2-1/2 to 3 times as great as that atproportional limit in compression parallel to grain. This means, then, that theouter transverse compression ply of a 3-ply plywood strip with all plies the samethickness reaches its proportional limit at about the same time as does thelongitudinal ply. The same will be true of plywood with more than three plies.Therefore, the outermost longitudinal ply will be the controlling factor, and theproportional limit of the plywood strip will be reached when the stress in thisply reaches its proportional limit value.

From the above considerations, calculations of F2 have been made on the

basis that the compressive transverse ply is effective, but that the tensiletransverse ply is completely ineffective. Under this assumption and neglectingthe effect of change of neutral axis when the outer transverse tension ply isneglected, equation (5) becomes

(9)

where E ' is the apparent modulus of elasticity about the centroidal axis with the2

outer transverse tension ply neglected and c' is the distance from the centroidal

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axis to the outer longitudinal ply. If all plies are of equal thickness, then

(10)

where n is the number of plies. Making this substitution into equation (9), equa-tion (5) becomes

(11)

Based on this equation the apparent stress at proportional limit for plywood withouter-perpendicular plies is

(12)

and for different number of plies, F2 can be calculated from the followingequations:

Number Equation for E2' Equation for F

2ofplies

1 +3 E

2' =

2713r

ELF = 1 + 13r FL2 9

26 +5 E2' =

12550r

EL

F = 26 + 50r FL2 75

+7 E' = 99 + 135rEL

F2

= 99

245

135r FL2 343

9 E2' = 244 + 292r EL

F = 244 + 292rFL729 2 567

Values of F2 calculated on the basisofthese equations are shown in column 17

of tables 5, 6, and 7. Values of F2

were obtained by use of equation (6) from data

on outer-perpendicular specimens and are shown in column 18 of the same tables

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as F2 (obs), The ratios of observed to calculated values of F2 are shown in

column 22 of tables 5, 6, and 7 and are summarized in table 9. While consider-able variability is found, the average value of the ratio is found to be about93 percent.

From the foregoing, it may be concluded that the load-carrying capacity (forstresses at the proportional limit) of a strip of plywood may be predicted by theequation

(13)

where K is an empirical factor, I is the moment of inertia of the whole crosssection, c is the distance from the centroidal axis to the extreme fiber, and F

amay be found from the following tabulation forplywood with all plies of the samethickness and species.

Number Equation for Fof a

pliesOuter-parallel Outer-perpendicular

r + 26 13r + 13 F1

= FL 27 F 2 =FL 9

50r + 265 F1 = FL26 + 99

125F2 = F

L 75

7 99r + 244F =F 135r + 99

F1 = FL

343 2 L 245

244r + 405 292r + 2449 F1 = FL 729 F2 =FL 567

A method is given elsewhere in this reportfor setting up similar formulas forany construction.

Modulus of Rupture

Obviously, any theories based upon assumptions of elastic behavior are invalidfor application at failure. However, it is not unreasonable to assume that values

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of apparent modulus of rupture for plywood, based upon the theory outlined in thepreceding section, would differ from the observed values by some approximatelyconstant factor.

Upon this premise, therefore, values of apparent modulus of rupture basedupon moduli determined from tests of all-parallel and all-perpendicular speci-mens were calculated on the same basis as that employed in calculating valuesof apparent fiber stress at proportional limit. These calculated values were com-pared with observed values of apparent modulus of rupture from tests of outer-parallel and outer-perpendicular specimens,

Moduli of rupture from tests of all-parallel and all-perpendicular specimensare shown in tables 5, 6, and 7 in columns 6 and 11 as SL and S T , respectively.

Values of S and S calculated from these moduli are shown in columns 8 and 13,1 2

respectively. Values of S2

were calculated assuming outer tension ply ineffective

as was explained for the calculation of F 2. Observed values of S 1 and S2 as found

from data of outer-parallel specimens and outer-perpendicular specimens areshown in columns 16 and 19, respectively. The ratios of observed to calculatedvalues are shown in columns 21 and 23 of tables 5, 6, and 7.

It may be concluded that the ultimate strength of a strip of plywood may bepredicted by the equation

(14)

where S may be found from the followingtabulation for plywood with all plies ofathe same thickness and species, and the other terms have the meanings given inconnection with equation (13)

Number Equation for Sof a

plies Outer-parallel Outer-perpendicular

13r + 13

S1 = SL 27S

2= S

L 9r + 26

5

7

9

26r + 99S1 = S

L 12550r + 26

S2

= SL 75

99r + 244 135r + 99S

2= S

L 245S1

= SL

343

244r + 485 292r + 244S

1= S

L729

S2 = SL 567

FPL-059 -16-

A later portion of this report will give a method of setting up similar equationsfor any construction.

Tables 8 and 9 present summaries of the comparisons between observed andcalculated values given in detail in tables 5, 6, and 7. In table 8, the values ofS

1ratios for all numbers of plies are considered together. The average ratio of

S1

(obs) to S1

(calc.) is 0.86. In table 9, however, the value of S2

ratios for the 5-,

7-, and 9-ply constructions are considered together in one group and those forthe 3-ply are considered separately, since the average ratio for 3-ply construc-tions is 1.15, and for the 5-, 7-, and 9-ply is approximately 1.0.

Effect of Neglecting Transverse Plies

Consideration of tables 1, 2, and 3 and 5, 6, and 7 indicates that the transverseplies may be expected to contribute, ingeneral, but little to the strength or stiff-ness of a plywood strip. A comparison of values of modulus of elasticity fromtests of all-perpendicular specimens with those from tests of all-parallel speci-mens shows that E

Tis only about 3 to 5 percent of E

L. Similarly, a comparison

of moduli of rupture indicates that the ultimate strength in tangential direction isonly about 4 to 6 percent of that in the longitudinal direction.

From this it would appear that a good approximation could be obtained byneglecting the effect of the transverse plies and by considering that the plywoodconsisted only of the longitudinal plies, with the transverse plies acting only tospace the longitudinal plies. Computations of modulus of elasticity, fiber stressat proportional limit, and modulus of rupture were made on this basis by con-sidering only those plies with the grain direction parallel to the span.

Modulus of Elasticity

In the case of modulus of elasticity, neglecting the effect of the transverseplies amounts, essentially, to neglecting the terms involving r (r = ET /EL ) in

equations (3). The errors to be expected from this approximation may be esti-mated. Assuming r = 0.04 for the 3-ply outer-parallel construction, for instance,the ratio of exact to approximate value would be 26.04/26 = 1.0015. The error isonly a small fraction of 1 percent. Other values of the ratio of exact to approxi-mate values of E1 and E 2 are tabulated below for r = 0.04.

FPL-059 -17-

Number Ratio of exact to approximateof modulus of elasticity

pliesOuter-parallel

(E1)

3 1.0015

5 1.0105

7 1.0162

9 1.0201

Outer-perpendicular(E2)

2.0400

1.1523

1.0986

1.0795

For the outer-parallel specimens, the error is negligible in comparison withthe natural variability of the material. For the outer-perpendicular specimens,on the other hand, the errors, particularly for the 3-ply, are of significant size.

Tables 10, 11, and 12 present values of EL

, FL

, and SL

calculated from the

results of tests of plywood specimens by considering only the parallel plies. Incolumns 6 and 7 are shown ratios of the values of E from such calculations

Lcompared with results from tests of all-parallel specimens. The ratios aregenerally near unity except for those for the 3-ply outer-perpendicular specimens.

As previously pointed out, approximate values of E may be calculated byc

neglecting the terms involving r in equations (3). The ratios of values calculatedfrom equations (3) considering the terms involving r to those calculated by neg-lecting such terms are shown in table 13 for various values of r. The values of rchosen are average values (table 16) for the various groups of specimens. Theratios should correspond with those given in columns 6 and 7 of tables 10, 11,and 12. The comparison, shown in table 14, indicates good agreement.

Figures 5 (left) and 6 (top) show the variation in error to be expected fromneglecting the transverse plies for various values of r and for 3-, 5-, 7-, and9-ply plywood. Figure 5 shows this relation for plywood with all plies the samethickness, and figure 6 for plywood with the face plies one-half as thick as theothers.

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Fiber Stress at Proportional Limit;Modulus of Rupture

Consider equation (5):

(5)

For a 3-ply strip with the grain of outer plies parallel to the span, this equationreduces to

(15)

for stress in the extreme fiber.

Neglecting the effect of the transverse plies here again amounts essentially toneglecting r in equation (15) and in the related equations for other constructions.The errors introduced by this approximation are indicated in figures 5 (right)and 6 (bottom) and by the ratios shown incolumns 11, 12, 16, and 17 of tables 10,11, and 12.

For outer-parallel specimens, the error resulting from neglecting the trans-verse plies in computing modulus of elasticity, fiber stress at proportional limit,or modulus of rupture is so small as to be negligible, even for plywood with alarge number of plies. For outer-perpendicular specimens, on the other hand,the error is considerable, especially for 3-ply and 5-ply plywood. For such con-structions, the approximation may be too greatly in error to be useful.

Relation Between the Properties of

Laminated Wood and Solid Wood

In the preceding sections, certain relationships between the properties oflaminated wood and plywood have been developed with the tacit assumption thatthe properties of laminated wood and solid wood were identical. Some consider-ation must be given to the validity of this assumption.

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In general, there appeared to be no variation in the strength properties of thelaminated wood with variations in ply thickness or in number of plies, exceptthat the results from tests of single veneers were generally somewhat lowerthan those from specimens withmorethanoneply. The reason for this differenceis not known, but it is possible that the glue joints in the multi-ply specimensmay have had some effect in increasing the strength,

Bending tests of solid wood have indicated a decrease in both fiber stress atproportional limit and modulus of rupture with increasing depth of beam (10). Itmight have been expected, then, that a tendency toward decreasing strength wouldbe found for laminated material with increasing numbers of plies of a givenveneer thickness, or with increasing ply thickness for specimens of a givennumber of plies. Such a trend was not found in either case. The variability ofresults was considerable, and apparently was sufficiently large to obscure atrend if any were present. The comparisonof specimens of a given veneer thick-ness for various numbers of plies would be expected to give the better compari-son, since such specimens would have come more nearly from the same portionof the tree than would those with the same number of plies but of different veneerthicknesses.

Obviously, any exact matching of material in laminated and solid specimens isimpossible. It was considered, however, that some semblance of matching mightbe obtained by testing solid specimens from the cores of the peeler blocks fromwhich the veneer had been cut. The results of standard bending tests on speci-mens from peeler block cores are summarized in table 15. The results, adjustedto a moisture content basis of 10 percent, are also shown to facilitate compari-son with the results of the tests of laminated specimens. Specific gravity valueswere not adjusted to the 10 percent moisture content basis.

Because of the limitations of size of available material, bending tests on speci-mens of solid wood with the grain perpendicular to the span could not be made.

The results from the tests of solid specimens bore no constant relation to theresults from the tests of laminated specimens, although the averages were gen-erally within the range of results from laminated specimens. It is probable thatthe matching was inadequate, because the solid specimens came from near thecenter of the tree and the laminated specimens came from varying positions nearthe outside. The general agreement of results, however, indicates that there isno serious difference in properties between solid and laminated specimens, andthat the calculation of plywood properties using the methods discussed herein,together with strength properties from tests of solid wood, is valid.

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Conclusions and Recommendations

The preceding portion of the report has presented methods of computing thestrength and stiffness of plywood strips subjected to bending, with a discussionof the relationships existing between the methods. A summary of these methodswill now be presented, together with the factors to be used in applying them.These factors apply to plywood having all equal thickness plies.

Deflection of Plywood Strips

Exact method.--The deflection of a plywood strip may be determined (exclusiveof deflections resulting from shear deformations) by use of the usual equations,simply by replacing modulus of elasticity in bending by an apparent modulus.This apparent modulus of elasticity in bending is a function of the number andthickness of the plies in the strip, and of the longitudinal and transverse moduliof elasticity of the wood.

The apparent modulus of elasticity, E , may be calculated fromc

where I is the moment of inertia of the full cross section about its central line,E.i the modulus of elasticity of the ith ply parallel to the span, Ii the moment of

inertia of the ith ply about the central line of the full cross section, and n thenumber of plies.

For plywood with all plies of the same species, E may be found fromc

where I is the moment of inertia of the full cross section about its central line,I and I are the moments of inertial of the transverse and longitudinal plies,x wrespectively, about the central line of the full cross section, r is the ratio ofthe transverse to longitudinal modulus of elasticity of the wood, and E L is the

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longitudinal modulus of elasticity of the wood (9 ). A table of values of E forc

plywood with all plies of the same species and the same thickness has been given,equation (3).

Approximate method. --The deflection may be computed approximately, usinga moment of inertia based upon only the parallel plies; that is,

where the summation is made only for those plies with the grain direction parallelto the span. For plywood with all plies of the same species,

For plywood with the face grain parallel to the span, this approximation will, formost cases, be only slightly in error. For plywood with the face grain perpendic-ular to the span, however, the error may be considerable and in many cases willbe so large as to render the approximation unusable. The magnitude of the errorsto be expected from the approximation has been shown, for certain cases, infigures 5 (left) and 6 (top).

Values of E (modulus of elasticity in the tangentialdirection of the wood) andT

E (modulus of elasticity in the radial direction of the wood) for computing theR

ratios r = E /E are available for some species. The dataT T L

and rR

= ER /EL

are incomplete, however, in that only a few species have been tested, and littleis known about the variation of these ratios with variations in specific gravity orwith changes in moisture content.

The principal available data are presented in “Elastic Properties of Wood,”Forest Products Laboratory Report No. 1528, and supplements thereto (4). Somedata are available also in references (6) and (8). For cases in which no data areavailable, the use of values of r = 0.10 is suggested. Approxi-= 0.05 and r

T Rmate values of these ratios may give results only slightly in error.

The correction for anticlastic curvature is usually so small that it may beneglected. From table II of March’s paper (7), it may be seen to be less than2 percent for 3-ply and 5-ply strips of spruce plywood, even for strips with aratio of span to width as low as 4.

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The correction for shear deformation, on the other hand, may be considerable.This correction is made (see equation (1)) by multiplying by a factor involvingthe constant e, which depends upon Poisson's ratios and moduli of rigidity. Dataon these properties are available (4, 6) for a few species, and may be used withequations presented by March to calculate values of e. Some values of this con-stant have already been calculated and are presented in table 18.

These values of e may be used as a basis for estimating values of e for otherspecies. Reasonable corrections for shear deformations may be made even withvalues of e considerably in error. Assume, for example, that the proper shearcorrrection is (1 + 0.10) and that the value of e is 20 percent low. The computedcorrection would then be (1 + 0.08), which is only 2 percent in error, so that thecorrected deflection would be only about 2 percent in error.

Load-Carrying. Capacity of Plywood Strips(For stresses at proportional limit)

Exact method.--The load-carrying capacity of a plywood strip for stresses atthe proportional limit may be determined by

where K is an empirical factor, F is found as indicated below, I is the momenta

of inertia of the whole cross section about its central line, and c is the distancefrom the centrodial axis to the extreme fiber of the outermost longitudinal ply.

The term F for outer-parallel strips is given bya

where F is an apparent stress in bending, FLis the desired stress in the outer-

amost longitudinal ply, and the other terms are as previously defined.

For outer-parallel strips with all plies of the same species, F may be founda

from

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where r , I , and I are as previously defined, and F is the desired stress inx w L

the outermost longitudinal ply.

For outer-perpendicular strips, F is given by a similar procedure, excepta

that the outer transverse ply on the tension side will be neglected. Here, Fa

will be given by

where E equals E for a strip ofplywood with the outer plies at right angles tom c

the span, and the outer ply on the tension side is considered ineffective. The otherterms are as previously defined.

For outer-perpendicular strips with all plies of the same species, F will bea

given by

where I' is the moment of inertia about the central line of the full cross sectionx

of all transverse plies except the one on the tension side, and all other symbolshave the meanings previously given.

Omission of the outer transverse ply on the tension side will result in a shiftin the neutral axis from the center of the depth toward the compression face ofthe beam. The effect of this shift is generally small enough to be neglected, andthe neutral axis may be assumed to be at the center of the depth of the beam.

The factor K has been determined experimentally and the following values will,in general, give results on the safe side: outer-parallel plywood, 0.85; outer-perpendicular plywood, 0.90.

Approximate method. --The load-carrying capacity may be computed approxi-mately by

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where the terms are as previously defined. The same values of the factor K maybe used, except for the 3-ply outer-perpendicular plywood, where a value of 1.20should be used,

For plywood with the face grain parallel to the span, this approximation, whichinvolves neglecting the transverse plies, will, for most cases, be only slightly inerror. For plywood with the face grain perpendicular to the span, however, theerror may be considerable and in many cases will be so large as to render theapproximation unusable, The magnitude of the errors to be expected from the ap-proximation has been shown, for certain cases, in figures 5 (right) and 6 (bottom).

Ultimate Load-Carrying Capacityof Plywood Strips

Exact method.--The ultimate load-carrying capacity of a plywood strip may becomputed by the method given for stresses at the proportional limit by substitu-ting modulus of rupture in place of the lower stress and by using appropriatevalues of K, which are somewhat different for ultimate load than for the lowerstresses.

Values of K are: outer-parallel plywood, 0.85; outer-perpendicular plywood,3-ply, 1.15; outer-perpendicular plywood, 5 plies or more, 1.00.

Approximate method.--The approximate method, as given for use with stressesat the proportional limit, may be used also for computing the ultimate load-carrying capacity by substituting modulus of rupture in place of the lower stressand by using appropriate values of K. Values of K given for use with the exactmethod may be used with the approximate method except for 3-ply outer-perpendicular plywood, where a value of 1.50 should be used.

Literature Cited

1. American Society for Testing Materials.1946. Standard methods of static tests of timber in structural sizes.

ASTM Designation D198-27.

2. .1946. Standard methods of testing small clear specimens of timber.

ASTM Designation D143-27.

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3. Bechtel, S. C., and Norris, C. B.1952, Strength of wood beams of rectangular cross section as affected

by span-depth ratio. Forest Products Lab. Rpt. 1910.

4. Doyle, D. V,, Drow, J. T., and McBurney, R. S.1945-46. Elastic properties of wood. Forest Products Lab. Rpt. 1528

and Supplements A, B, C, D, F, G, H.

5. Drow, J. T.1945. Effect of moisture on the compressive, bending, and shear

strengths, and on the toughness of plywood. Forest ProductsLab. Rpt. 1519.

6. Jenkin, C. F.1920. Report on materials of construction used in aircraft and aircraft

engines. (Gt. Brit.) Min. Munitions, Aircraft Production Dept.,Aeronautical Research Com.

7. March, H. W.1936. Bending of a centrally-loaded rectangular strip of plywood.

Physics 7(1):32-41.

8. Markwardt, L. J.1938. Form factors and methods of calculating the strength of a wooden

beam, Forest Products Lab. Rpt. 1184.

9. , and Wilson, T. R. C.1935. Strength and related properties of woods grown in the United

States. U.S. Dept. Agr. Tech. Bul. 479.

10. Newlin, J. A., and Trayer, G. W.1924. The influence of the form of a wooden beam on its stiffness and

strength, II. Form factors of beams subjected to transverseloading only. Natl. Advisory Comm. for Aeronautics Rpt. 181.

11. Norris, C. B.1942. The technique of plywood. I. F. Laucks, Inc., Seattle.

12. Wilson, T. R C.1932. Strength-moisture relations for wood. U.S. Dept. Agr. Tech.

Bul. 282.

FPL-059 -26- 1.-56

Table 1.--Comparison of observed and calculated values of E1 and E2 for Sitka spruce. (Allvalues except specific gravity adjusted to 10 percent moisture content basis.)

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Table 1.--Comparison of observed and calculated values of E1 and E2 for Sitka sprucde. (All values

except specifice gravity adjusted to 10 percent moisture content basis.) (continued)

(Sheet 2 of 2)

-28-

Table 2.-- Comparison of observed and calculated values of E1 and E2 for Douglas-fir. (All

values except specific gravity adjusted to 10 percent moisture content basis.)

: 1,539 :: 1,115 :: 1,327 :

: 1,489 :: 1,121 :: 1,305 :

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Table 2.--Comparison of observed and calculated values of E1 and E2 for Douglas-fir. (All values

except specific gravity adjusted to 10 percent moisture content basis.) (continued)

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Table 3.--Comparison of observed and calculated values of E1 and E2 for yellow-poplar. (Allvalues except specific gravity adjusted to 10 percent moisture content basis.)

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Table 4,--Summary of ratios of observed to calculated values from table 1

Table 5.--Comparison of observed and calculated values of F1, S1, F2, and S2 for Sitka spruce.

(All values except specific gravity adjusted to 10 percent moisture content basis.)

(Sheet 1 of 2)

Table 5.--Comparison of observed and calculated values of F1, S1, F2, and S2 for Sitka spruce. (All

except gravity adjusted to percent moisture content basis.) (continued)

2 Of 2)

Table 6.--Comparison of observed and calculated values of F1, S1, F2 and S2 for Douglas-fir.

(All values except specifice gravity adjusted to 10 percent moisture content basis.)

(Sheet 1 of 2)

Table 6.--Comparison of observed and calculated values of F1, S1, F2, and S2 for Douglas-fir. (All

values except specific gravity adjusted to 10 percent moisture content basis.) (continued)

(Sheet 2 of 2)

FP

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812–915–4

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Table 7.--Comparison of observed and calculated values of F1, S1, F2, and S2 forf yellow-poplar.

(All value except specific gravity adjusted to 10 percent moisture content basis.)

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Table 8.-- Summary of ratios of observed to calculated values of F1 and S1 from tables 5, 6, and 7

Table 9.-- Summary of ratios of observed to calculated values of F2 and S2 from tables 5, 6, and 7

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Table 10.--Comparison of values of modulus of elasticity, fiber stress at ID rtional limit andmodulus of rupture from tests of all-parallel, outer-parallel, and outer-perpenmarspecimens, neglecting effect of transverse plies (Sitka spruce) (continued)

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Figure 2.--Details of apparatus for bending tests of plywood, showing method of attachment ofdial gage to specimens too short to permit use of supporting yoke as shown in figure 1.

Figure 3.--Details of apparatus for bending tests of plywood, showing the use of a platform scalefor measuring small loads.

GPO 812–915–2

Figure 4.--Details of apparatus for bending tests of single-ply veneer.

Figure 6.--Ratios of deflections (top) and moments (bottom) calculated by neglecting transverseplies to deflections calculated by considering transverse plies when the outer plies are one-half as thick as the other plies.

Z M 41128 F